Quantum state discrimination

Abstract

It is a fundamental consequence of the superposition principle for quantum states that there must exist non-orthogonal states, that is states that, although different, have a non-zero overlap. This finite overlap means that there is no way of determining with certainty in which of two such states a given physical system has been prepared. We review the various strategies that have been devised to discriminate optimally between non-orthogonal states and some of the optical experiments that have been performed to realise these.

Full text

Page 1

arXiv:0810.1970v1 [quant-ph] 10 Oct 2008
Quantum state discrimination
Stephen M. Barnett1,∗and Sarah Croke2
1Department of Physics, University of Strathclyde
Glasgow G4 0NG, UK
2Perimeter Institute for Theoretical Physics
Waterloo, Ontario, N2L 2Y5, Canada
∗Corresponding author: steve@phys.strath.ac.uk
It is a fundamental consequence of the superposition principle
for quantum states that there must exist non-orthogonal states,
that is states that, although different, have a non-zero overlap.
This finite overlap means that there is no way of determining
with certainty in which of two such states a given physical
system has been prepared. We review the various strategies
that have been devised to discriminate optimally between
non-orthogonal states and some of the optical experiments that
have been performed to realise these. © 2008 Optical Society of
America
OCIS codes: 270.5565,270.5585.
1.Introduction
The state of a quantum system is a mysterious object and has been the subject
of much attention since the earliest days of quantum theory. We know that it
provides a way of calculating the observed statistical properties of any desired
observable but that it is not, itself observable. This means that we cannot deter-
mine by observation the state of any single physical system. If we have some
prior information, however, then we may be able to use this to determine, at
least to some extent, the state. Consider, for example, a single photon which
we know has been prepared with either horizontal or vertical polarisation. A
suitably oriented polarising beam splitter can be used to transmit the photon if
it was vertically polarised and reflect it if its polarisation was horizontal. De-
termining the path of the photon by absorbing it with a suitable detector then
determines the state to have been one of horizontal or vertical polarisation.
Suppose, however, that we are told that our photon was prepared with either
horizontal or with left-circular polarisation. These quantum states of polarisa-
tion are not orthogonal in that states of circular polarisation are superpositions

Page 2

of those of both vertical and horizontal polarisation:
|L? =
1
√2(|H?+i|V?)
1
√2
⇒ ?H|L? =
?= 0.
(1)
If we subject our photon to the polarisation measurement outlined in the pre-
ceding paragraph then a left-circularly polarised photon will appear to be hor-
izontally polarised with probability1
probability.
The problem of discriminating between such states is fundamental to the
quantum theory of communications [1–4] and underlies the secrecy of the now
well-reviewed science of quantum cryptography [5–9]. Indeed, we can use the
connection between quantum state discrimination and quantum communica-
tions to motivate the problem of state discrimination. We suppose that two
parties, conventionally named Alice and Bob, wish to communicate using a
quantum channel. To do this Alice selects from a given set of states, |ψi? (or
more generally mixed states with density operators ˆ ρi) with a given set of prob-
abilities pi. The selected state is encoded in the preparation of a given physical
system, such as photon polarisation, and this is sent to Bob. Bob will know both
the set of possible states and the associated preparation probabilities. His task
is to determine, by means of a suitable measurement, the state selected by Alice
and hence the intended message. This, then is the quantum state discrimination
problem: how can we best discriminate among a known set of possible states
|ψi?, each having been prepared with a known probability pi.
The quantum state discrimination problem, as posed here, has been the sub-
ject of active theoretical investigation for a long time [1–3,10–14], but is only
comparativelyrecently that experimentshavebeen performed and mostof these
have been based on optics. There exist in the literature a number of reviews of
and introductions to quantum state discrimination [4,15–20]. Our purpose in
preparing this review is twofold: first to bring the rapidly developing field up
to date and secondly to introduce the idea of state discrimination to a wider
audience in optics. It seems especially appropriate to do this as it is in simple
optical experiments that the ideas are most transparent and where most of the
important practical developments have been made.
2and vertically polarised with the same
2.Generalised measurements
Most of us are introduced to the idea of measurements in quantum theory in a
manner that is, essentially, that formulated by von Neumann [21]. Each observ-
able property O is associated with a Hermitian operatorˆO (or more precisely a

Page 3

self-adjoint one) the eigenvalues of which are the possible results of a measure-
ment of O. If the eigenvalues and eigenvectors are omand |om? then we can
write the operatorˆO in the diagonal form
ˆO =∑
m
om|om??om|.
(2)
If the system to be measured has been prepared in the state |ψ? then the proba-
bility that a measurement of O will give the result omis
P(om) = |?om|ψ?|2.
(3)
Consider, for example, a measurement to determine whether the polarisation of
a single photon is horizontal or vertical. A suitable operator, corresponding to
this measurement, would be
ˆ Pol = H|H??H|+V|V??V|.
(4)
The probability that a measurement of this property on a photon prepared in the
circularly polarised state |L? will give the result H, corresponding to horizontal
polarisation, is
P(H) = |?H|L?|2=1
It is helpful, in what follows, to rewrite the above probabilities as the expec-
tation value of an operator. In this way the probability that a measurement of
optical polarisation shows the photon to be horizontally polarised is
2.
(5)
P(H) = ?|H??H|? = ?ˆPH?,
(6)
whereˆPH= |H??H|, the projector onto the state |H?, is the required operator.
More generally, for our operatorˆO, the probability that a measurement gives
the value omis
P(om) = ?|om??om|? = ?ˆPm?.
We note that the measured value, itself, makes no explicit appearance in this
probability; it is not the eigenvalue but only the corresponding eigenvector that
determinestheform oftheprojectorandhence theprobabilityfortheassociated
measurement outcome.
The projectors have four important mathematical properties and it is helpful
to list these:
(7)
• The projectors are Hermitian operators,ˆP†
ated withthefact that probabilitiesare, themselves,observablequantities.
m=ˆPm. This property is associ-

Page 4

• They are positive operators, which means that ?ψ|ˆPm|ψ? ≥ 0 for all pos-
sible states |ψ?. This reflects the fact that the expectation value of the
projector is a probability and must, therefore, be positive or zero.
• They are complete in that ∑mˆPm=ˆ
for all possible measurement outcomes is unity.
?, so that the sum of the probabilities
• They are orthonormal in thatˆPmˆPn= 0 unless m = n. This property is
sometimes associated with the fact that measurement outcomes must be
distinct (you can only get one of them). This view is, as we shall see, not
correct. You can indeed only get one outcome but this does not require
the orthonormality property.
The theory of generalised measurements can be formulated simply by drop-
ping the final requirement. To see how this works, we introduce a set of proba-
bility operators { ˆ πm}, each of which we wish to associate with a measurement
outcome such that the probability that our measurement gives the result labeled
m is
P(m) = ? ˆ πm?.
We insist on the first three of the properties of the projectors, as these are re-
quired if we are to maintain the probability interpretation (8), but drop the final
requirement so that our probability operators have the properties:
(8)
• The probability operators are Hermitian: ˆ π†
m= ˆ πm.
• They are positive operators: ?ψ|ˆ πm|ψ? ≥ 0 for all possible states |ψ?.
• They are complete: ∑mˆ πm=ˆ
?.
The set of probability operators characterising the possible outcomes of any
generalised measurement is called a probability operator measure, usually ab-
breviated to POM [1,4]. You will often find this set referred to as a positive
operator-valued measure or POVM [22,23]. If the latter name is used then the
probability operators become elements of a POVM.
The differences between the projectors and more general probability opera-
tors are best appreciated by reference to some simple examples and these will
be given in the followingsections. There are, however, some important and per-
haps even surprising points and it is sensible to emphasise these here. Firstly,
the three properties described above have a remarkable generality in that (i)
any measurement can be described by the appropriate set of probability oper-
ators and (ii) any set of operators that satisfy the three properties correspond
to a possible measurement [4,22]. This means that we can seek the optimum

Page 5

measurementinanygivensituationmathematically,bysearchingamongallsets
of operators that satisfythe required properties. Having found thiswe knowthat
a physical realisation of this will exist and can seek a way to implement it in the
laboratory. The second point to emphasise is that the number of (orthogonal)
projectors can only be less than or equal to the dimension of the state space.
For optical polarisation, for example, there are only two orthogonal polarisa-
tions and the state space is therefore two-dimensional. It follows that any von
Neumann measurement of polarisation can only have two outcomes. By drop-
ping the requirement for orthogonality, we allow a generalised measurement
to have any number of outcomes, so a generalised measurement of polarisa-
tion can have three, four or more different outcomes. Finally, a generalised
measurement allows us to describe the simultaneous observation of incompati-
ble observables, such as position and momentum or, in the context of quantum
optics, orthogonal field quadratures [24,25]. Perhaps the first reported gener-
alised optical measurement was of precisely this form [26,27].
3. State Discrimination -Theory
3.1.Mimimum Error Discrimination
In quantumstatediscriminationwewishto designameasurementtodistinguish
optimallybetween a givenset of states.As wehaveseen in theprevioussection,
any physically realisable measurement can be described by a probability oper-
ator measure. Thus by mathematically formulating a figure of merit describing
the performance of a measurement, we can search for the set of probability op-
erators describing the optimal measurement. There are several possible figures
of merit, each one corresponding to a different strategy. Possibly the simplest
criteria which may be applied is to minimise the probability of making an error
in identifying the state. We begin with the special case where the state is known
to be one of two possible pure states, |ψ0?, |ψ1?, with associated probabilities
p0, p1= 1− p0. If outcome 0, associated with the probability operator ˆ π0is
taken to indicate that the state was |ψ0?, and outcome 1 (associated with ˆ π1) is
taken to indicate that the state was |ψ1?, the probability of making an error in
determining the state is given by
Perr = P(ψ0)P(1|ψ0)+P(ψ1)P(0|ψ1)
=
p0?ψ0|ˆ π1|ψ0?+ p1?ψ1| ˆ π0|ψ1?
=
p0−Tr((p0|ψ0??ψ0|− p1|ψ1??ψ1|)ˆ π0),
where in the last line we have used the completeness condition ˆ π0+ ˆ π1=ˆ
This expression takes its minimum value when the second term reaches a maxi-
mum, which in turn is achieved if ˆ π0is a projector onto the positive eigenket of
(9)
?.